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When ice melts in a full glass of water, will the water overflow

  1. Mar 21, 2012 #1
    If you have an ice cube in a full glass of water when the ice cube melts will the water flow over, be the same, or decrease? My book says the answer is stay the same, but I can't figure out why. Ice was less dense than water which is why is floats. Using this equation:

    B = ρVg which when solved for V becomes

    B/(ρg) = V

    I would think the buoyancy would stay the same as the ice cube melts so the density ρ increases as the ice melts, so if you increase the denominator, the whole number decreases. So the water should decrease, not stay the same, unless the increase in density is compensated for by an increase in Buoyancy.
  2. jcsd
  3. Mar 21, 2012 #2


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    I shouldn't just hand this to you, I should walk you through it, but you're made some effort - and the diagram is right there...

  4. Mar 21, 2012 #3


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    An also interesting question would be why the melting of icebergs increase the level of oceans (I think it does, but I could be wrong).
  5. Mar 21, 2012 #4
    Ok, I'm using this equation now

    ρwaterAh = ρcubeVcube

    As the volume decreases so too does the area and the height.
  6. Mar 21, 2012 #5


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    As already posted the same.

    One somewhat novel exception is that that if you had a higher density form of ice, such as ice III (or higher stage of ice), it would be resting on the bottom of the glass instead of floating. In this case, as it melts, the density would decrease and the water level would rise.

    It's the melting of glaciers which are currently supported by land masses, not the oceans. As they melt, the water eventually runs off into the oceans.
  7. Mar 21, 2012 #6


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    Ok thanks rcgldr!
  8. Mar 21, 2012 #7
    If the ice cube was chained to the base of the glass, then as it melts, it turns into denser water, so the water level will lower.

    If the ice cube is buoyant, then the amount of water it displaces is equal to its own WEIGHT (not mass as someone else wrote); as it melts, the water level remains the same. The ice cube is less dense than water, so it will be "sticking up" above the water, and as it melts, it turns into denser water, occupying the same volume as the submerged portion of the ice cube.

    If the ice cube had helium bubbles in it, then the helium would add even greater buoyancy, as it has negative weight against the atmosphere, and as the helium leaked out, the ice cube would WEIGH more (even though, without the helium bubbles, it has less mass), and so the water level will rise.

    It is important to distinguish weight and mass. There's only a notion of "displacing" water because there's a force applied on the water to _prevent_ it from natural displacement (brownian motion). In this case, gravity and the contours of the glass. The water has a downward force keeping it from naturally flying out of the glass (gravity) - a weigh scale can measure this force. A buoyant object also applies a downward force - again measurable by a weigh scale. The water moves /upward/ in the presence of the buoyant object trying to go /downward/ because of the contours of the glass. If the glass were a pipe, you've just re-invented a pipe-cleaner and helped the liquid move further down the pipe. Because the glass has a strong bottom, the /bottom/ of the glass applies an upward force against the water. The water level then rises as the buoyant object submerges. If the only forces involved are gravity, then you'll get the same answer whether you use mass or weight, but if you introduce any other forces, the upward force on helium, or the downward force of a magnetic field tugging a buoyant object down, you'll get the incorrect answer using mass. For example, if you have an iron warship and a magnetic field pushing it downward, it will displace more water than an iron warship of equal mass without that additional downward force.
    Last edited: Mar 21, 2012
  9. Mar 21, 2012 #8
    Your diagram is sublime :D
  10. Mar 22, 2012 #9


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    Would it be mischievous to point out that the melting of the ice cube will change the temperature of the water, and hence its volume?

    Of course this effect will be temporary; eventually the whole thing returns to ambient temperature.
    Last edited: Mar 22, 2012
  11. Mar 22, 2012 #10
    Think of a boat filled with ice that's floating in a lake. The ice in the boat melts, but the mass doesn't change, and the waterline stays the same.
  12. Mar 22, 2012 #11


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    There are some modifications to that question:

    What happens to the water level during melting if the ice has...

    a) ... a cavity with air inside?

    b) ... a cavity with molten water inside?

    c) ... a piece of lead inside?
  13. Mar 22, 2012 #12


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    No, that is already covered by the temperature causing the water to be ice. Think of the water and ice as a closed system and consider the average temperature of this system.
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